Question

The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is $$q=f(k, l)=\left[(\alpha k)^{\rho}+(\beta l)^{\rho}\right]^{\gamma / \rho}$$ a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is \(v / \alpha\) and for a unit of labor input is \(w / \beta\) b. If \(\gamma=1\) and \(\alpha+\beta=1,\) it can be shown that this production function converges to the Cobb-Douglas form \(q=k^{\alpha} l^{\beta}\) as \(\rho \rightarrow 0 .\) What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by wl/ \(v k .\) Show that this share is constant for the Cobb-Douglas function in part (b). How is the relative labor share affected by the parameters \(\alpha\) and \(b ?\) d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in \(w / v ?\) How is the direction of this effect determined by the elasticity of substitution, \(\sigma\) ? How is it affected by the sizes of the parameters \(\alpha\) and \(\beta ?\)

Short answer

Question: Calculate the relative labor cost share for the general CES function. Answer: The relative labor cost share for the general CES function is: $$\frac{wl}{vk} = \frac{w \cdot \frac{[\frac{q^\rho}{\beta}]^{\frac{1}{\rho}} - \alpha k^{\rho}}{\beta}}{v \cdot \frac{[\frac{q^\rho}{\alpha}]^{\frac{1}{\rho}} - \beta l^{\rho}}{\alpha}}$$ The magnitude of this share depends on the ratio of \(w/v\), the elasticity of substitution (\(\sigma\)), and the values of \(\alpha\) and \(\beta\). If \(\rho > 0\), an increase in \(\sigma\) leads to a decrease in the labor cost share, while if \(\rho < 0\), an increase in \(\sigma\) results in an increase in the labor cost share. The values of \(\alpha\) and \(\beta\) influence the labor cost share similarly to the Cobb-Douglas case.

Step-by-step solution

Determine the expression for production function given in the problem.

The generalized CES production function has the form: $$q = [\alpha k^{\rho} + \beta l^{\rho}]^{\gamma / \rho}$$

Rewrite the production function by isolating the variables k and l.

Expressing k and l in terms of q and the other parameters: $$k = \left (\frac{[\frac{q^\rho}{\alpha}]^{\frac{1}{\rho}} - \beta l^{\rho}}{\alpha} \right)^{\frac{1}{\rho}}$$ $$l = \left (\frac{[\frac{q^\rho}{\beta}]^{\frac{1}{\rho}} - \alpha k^{\rho}}{\beta} \right)^{\frac{1}{\rho}}$$

Calculate the total cost function using given hint about prices.

The total cost function is expressed as \(C(w,v) = vk + wl\). Knowledge from Example 10.2 tells us that the price for a unit of capital input in this production function is \(v/\alpha\) and for a unit of labor input is \(w/\beta\). Now, we can substitute the values of k and l from the step above, and express the total cost function as: $$C(w,v) = w \cdot \frac{[\frac{q^\rho}{\beta}]^{\frac{1}{\rho}} - \alpha k^{\rho}}{\beta} + v \cdot \frac{[\frac{q^\rho}{\alpha}]^{\frac{1}{\rho}} - \beta l^{\rho}}{\alpha}$$ This is the total cost function for the firm with the generalized CES production function. b) Finding the total cost function for the specific version of the CES function.

Specify given conditions for the parameters.

We are given the conditions \(\gamma = 1\) and \(\alpha + \beta = 1\).

Find the CES production function under the specified conditions.

Under the specified conditions, the CES production function converges to the Cobb-Douglas form as \(\rho \rightarrow 0\): $$q = (\alpha k^{\rho} + \beta l^{\rho})^{\frac{1}{\rho}}$$ It converges to: $$q = k^{\alpha} l^{\beta}$$

Substitute the Cobb-Douglas function into the total cost function.

Substituting the values of k and l from the Cobb-Douglas function into the total cost function, we get: $$C(w,v) = w \cdot \frac{l}{\beta} + v \cdot \frac{k}{\alpha}$$ c) Show that the relative labor cost share is constant for the Cobb-Douglas function.

Calculate the relative labor cost share for the Cobb-Douglas function.

The relative labor cost share, represented by \(\frac{wl}{vk}\), can be found using the Cobb-Douglas total cost function: $$\frac{wl}{vk} = \frac{\frac{l}{\beta}}{\frac{k}{\alpha}}$$

Simplify the expression for the relative labor cost share.

After simplifying the expression, we find: $$\frac{wl}{vk} = \frac{l \cdot \alpha}{k \cdot \beta}$$ As this expression does not depend on l and k values, the relative labor cost share for this Cobb-Douglas function is constant.

Analyze the effect of \(\alpha\) and \(\beta\) on the labor cost share.

Increasing alpha raises the relative labor cost share, as l becomes more important. Similarly, increasing \(\beta\) lowers the labor cost share as the contribution of k becomes more important in production. d) Calculate the relative labor cost share for the general CES function.

Use the total cost function for the general CES function.

The relative labor cost share is expressed in the total cost function: $$\frac{wl}{vk} = \frac{w \cdot \frac{[\frac{q^\rho}{\beta}]^{\frac{1}{\rho}} - \alpha k^{\rho}}{\beta}}{v \cdot \frac{[\frac{q^\rho}{\alpha}]^{\frac{1}{\rho}} - \beta l^{\rho}}{\alpha}}$$

Analyze the effect of \(w/v\), \(\sigma\), \(\alpha\) and \(\beta\) on the labor cost share.

The magnitude of the relative labor cost share depends directly on the ratio of \(w/v\). The direction of the effect of the elasticity of substitution (\(\sigma\)) is determined by the sign of \(\rho\): if \(\rho > 0\), an increase in \(\sigma\) leads to a decrease in the labor cost share, and if \(\rho < 0\), an increase in \(\sigma\) leads to an increase in the labor cost share. The values of \(\alpha\) and \(\beta\) influence the labor cost share in a similar way as in the Cobb-Douglas case.